

Fan's trace min/max principle [1]} dictates that the solution to this minimization problem is given by a matrix \(\mathbf {U}\)  whose first \(k\)  columns are spanned by the eigenvectors associated with the \(k\)  smallest eigenvalues of \(\Delta _2\) .
In order to obtain discrete partitions from the resulting real valued eigenvectors we consider for the \(n\)  nodes of the graph \(n\)  vectors \(\mathbf {h}_i = \mathbf {U}^{{\mathchoice{\raisebox {}{\m@th \displaystyle \intercal }}{\raisebox {}{\m@th \textstyle \intercal }}{\raisebox {}{\m@th \scriptstyle \intercal }}{\raisebox {}{\m@th \scriptscriptstyle \intercal }}}}_i \in \mathbb {R}^k \; \forall i \in [1,n]\) . These are considered the spectral coordinates of the graph and are clustered into \(k\) -groups \(C_1, \dots , C_k\) . The performance of various clustering algorithms has been tested on the spectral coordinates, with k-means being considered the standard choice [2]}. The procedure of assigning nodes to these clusters is as follows:
\(v_i \in C_j \;\; \text{if} \; \mathbf {h}_i \in C_j \;\; \text{for}\; i = \lbrace 1,\dots ,n\rbrace , \, j = \lbrace 1,\dots ,k\rbrace .\) 
